Constructive Approximations of Markov Operators

We construct piecewise linear Markov finite approximations of Markov operators defined on L ¹([0, 1] ^N ) and we study various properties, such as consistency, stability, and convergence, for the purpose of numerical analysis of Markov operators.     

Matter wave interferometry for measuring a molecular transition dipole moment

We consider localized states of both single- and two-component Bose-Einstein condensates (BECs) confined in a potential resulting from the superposition of linear and nonlinear optical lattices and make use of Vakhitov-Kolokolov criterion to investigate the effect of nonlinear lattice on the stability of the soliton solutions in the linear optical lattice (LOL). For the single-component case we show that a weak nonlinear lattice has very little effect on the stability of such solitons while sufficiently strong nonlinear optical lattice (NOL) squeezes them to produce narrow bound states. For two-component condensates we find that when the strength of the NOL (γ ₁) is less than that of the LOL (V ₀) a relatively weak intra-atomic interaction (IAI) has little effect on the stability of the component solitons. This is true for both attractive and repulsive IAI. A strong attractive IAI, however, squeezes the BEC solitons while a similar repulsive IAI makes the component solitons wider. For γ ₁ > V ₀, only a strong attractive IAI squeezes the BEC solitons but the squeezing effect is less prominent than that found for γ ₁ < V ₀. We make useful checks on the results of our semianalytical stability analysis by solving the appropriate Gross-Pitaevskii equations numerically.     

Linear Perturbations of Quaternionic Metrics

We extend the twistor methods developed in our earlier work on linear deformations of hyperkähler manifolds [1] to the case of quaternionic-Kähler manifolds. Via Swann’s construction, deformations of a 4d-dimensional quaternionic-Kähler manifold ${\mathcal{M}}We extend the twistor methods developed in our earlier work on linear deformations of hyperk?hler manifolds [1] to the case of quaternionic-K?hler manifolds. Via Swann’s construction, deformations of a 4d-dimensional quaternionic-K?hler manifold M{\mathcal{M}} are in one-to-one correspondence with deformations of its 4d + 4-dimensional hyperk?hler cone S{\mathcal{S}}. The latter can be encoded in variations of the complex symplectomorphisms which relate different locally flat patches of the twistor space Z_S{\mathcal{Z}_\mathcal{S}}, with a suitable homogeneity condition that ensures that the hyperk?hler cone property is preserved. Equivalently, we show that the deformations of M{\mathcal{M}} can be encoded in variations of the complex contact transformations which relate different locally flat patches of the twistor space Z_M{\mathcal{Z}_\mathcal{M}} of M{\mathcal{M}}, by-passing the Swann bundle and its twistor space. We specialize these general results to the case of quaternionic-K?hler metrics with d + 1 commuting isometries, obtainable by the Legendre transform method, and linear deformations thereof. We illustrate our methods for the hypermultiplet moduli space in string theory compactifications at tree- and one-loop level.     

Multi-patch model for transport properties of cuprate superconductors

A number of normal state transport properties of cuprate superconductors are analyzed in detail using the Boltzmann equation. The momentum dependence of the electronic structure and the strong momentum anisotropy of the electronic scattering are included in a phenomenological way via a multi-patch model. The Brillouin zone and the Fermi surface are divided in regions where scattering between the electrons is strong and the Fermi velocity is low (hot patches) and in regions where the scattering is weak and the Fermi velocity is large (cold patches). We present several motivations for this phenomenology starting from various microscopic approaches. A solution of the Boltzmann equation in the case of N patches is obtained and an expression for the distribution function away from equilibrium is given. Within this framework, and limiting our analysis to the two patches case, the temperature dependence of resistivity, thermoelectric power, Hall angle, magnetoresistance and thermal Hall conductivity are studied in a systematic way analyzing the role of the patch geometry and the temperature dependence of the scattering rates. In the case of Bi-based cuprates, using ARPES data for the electronic structure, and assuming an inter-patch scattering between hot and cold states with a linear temperature dependence, a reasonable agreement with the available experiments is obtained. Received 3 August 2001 and Received in final form 1st November 2001     

Some Stability and Instability Criteria for Ideal Plane Flows

We investigate stability and instability of steady ideal plane flows for an arbitrary bounded domain. First, we obtain some general criteria for linear and nonlinear stability. Second, we find a sufficient condition for the existence of a growing mode to the linearized equation. Third, we construct a steady flow which is nonlinearly and linearly stable in the L² norm of vorticity but linearly unstable in the L² norm of velocity.     

Quasi-two-dimensional Bose–Einstein condensates with spatially modulated cubic–quintic nonlinearities

We investigate exact nonlinear matter wave functions with odd and even parities in the framework of quasi-two-dimensional Bose–Einstein condensates (BECs) with spatially modulated cubic–quintic nonlinearities and harmonic potential. The existence condition for these exact solutions requires that the minimum energy eigenvalue of the corresponding linear Schrödinger equation with harmonic potential is the cutoff value of the chemical potential λ. The competition between two-body and three-body interactions influences the energy of the localized state. For attractive two-body and three-body interactions, the larger the matter wave order number n, the larger the energy of the corresponding localized state. A linear stability analysis and direct simulations with initial white noise demonstrate that, for the same state (fixed n), increasing the number of atoms can add stability. A quasi-stable ground-state matter wave is also found for repulsive two-body and three-body interactions. We also discuss the experimental realization of these results in future experiments. These results are of particular significance to matter wave management in higher-dimensional BECs.     

Kink stability of isothermal spherical self-similar flow revisited

The problem of kink stability of isothermal spherical self-similar flow in newtonian gravity is revisited. Using distribution theory we first develop a general formula of perturbations, linear or non-linear, which consists of three sets of differential equations, one in each side of the sonic line and the other along it. By solving the equations along the sonic line we find explicitly the spectrum, k, of the perturbations, whereby we obtain the stability criterion for the self-similar solutions. When the solutions are smoothly across the sonic line, our results reduce to those of Ori and Piran. To show such obtained perturbations can be matched to the ones in the regions outside the sonic line, we study the linear perturbations in the external region of the sonic line (the ones in the internal region are identically zero), by taking the solutions obtained along the line as the boundary conditions. After properly imposing other boundary conditions at spatial infinity, we are able to show that linear perturbations, satisfying all the boundary conditions, exist and do not impose any additional conditions on k. As a result, the complete treatment of perturbations in the whole spacetime does not alter the spectrum obtained by considering the perturbations only along the sonic line.     

Stability of the Einstein static universe in IR modified Hořava gravity

Recently, Hořava proposed a power counting renormalizable theory for (3+1)-dimensional quantum gravity, which reduces to Einstein gravity with a non-vanishing cosmological constant in IR, but possesses improved UV behaviors. In this work, we analyze the stability of the Einstein static universe by considering linear homogeneous perturbations in the context of an IR modification of Hořava gravity, which implies a ‘soft’ breaking of the ‘detailed balance’ condition. The stability regions of the Einstein static universe is parameterized by the linear equation of state parameter w=p/ρ and the parameters appearing in the Hořava theory, and it is shown that a large class of stable solutions exists in the respective parameter space.     

Lagrangian relative equilibria for a gyrostat in the three-body problem

In this paper we consider the noncanonical Hamiltonian dynamics of a gyrostat in the three-body problem. By means of geometric mechanics methods, we study the approximate Poisson dynamics that arise when we develop the potential of the system in Legendre series and truncate this to an arbitrary order k. After reduction of the dynamics by means of the two symmetries of the system, we consider the existence and number of equilibria which we denominate of Lagrangian type, in analogy with classic results on the topic. Necessary and sufficient conditions are established for their existence in an approximate dynamics of order k, and explicit expressions for these equilibria are given, this being useful for the subsequent study of their stability. The number of Lagrangian equilibria is thoroughly studied in approximate dynamics of orders zero and one. The main result of this work indicates that the number of Lagrangian equilibria in an approximate dynamics of order k for k ≥1 is independent of the order of truncation of the potential, if the gyrostat S ₀ is almost spherical. In relation to the stability of these equilibria, necessary and sufficient conditions are given for linear stability of Lagrangian equilibria when the gyrostat is almost spherical. In this way, we generalize the classical results on equilibria of the three-body problem and many results provided by other authors using more classical techniques for the case of rigid bodies.      

A lattice hydrodynamical model considering turning capability

A new two-dimensional lattice hydrodynamic model considering the turning capability of cars is proposed. Based on this model, the stability condition for this new model is obtained by using linear stability analysis. Near the critical point, the modified KdV equation is deduced by using the nonlinear theory. The results of numerical simulation indicate that the critical point a c increases with the increase of the fraction p of northbound cars which continue to move along the positive y direction for c = 0.3, but decreases with the increase of p for c = 0.7. The results also indicate that the cars moving along only one direction (eastbound or northbound) are most stable.     

Analysis of domain-solutions in reaction-diffusion systems

We investigate a standard model for bistable reaction-diffusion-systems, which shares characteristic properties with the van-der-Pol oscillator for distributed generators and the FitzHugh-Nagumo system. In this system we study the effect of a long ranging inhibitor. As a main result we show the existence of two inhomogeneous stationary solutions—the smaller one is always a saddle which corresponds to a critical nucleus, while the larger one arises as astable solution. In carrying out the linear stability analysis for these solutions, we have to treat the Schrödinger-equation for a double-well potential. This is done approximately by a supersymmetric approach which yields the eigenvalues and eigenfunctions of the Schrödinger-equation. Furthermore we compare our analytical findings with numerical results-especially the occurrence of oscillating solutions is shown.     

Comment on “On the controversy around Daganzo’s requiem for and Aw-Rascle’s resurrection of second-order traffic flow models" by D. Helbing and A.F. Johansson

There has been a debate in the transportation research community over the validity of a class of traffic flow models. These models have the peculiar property that one of its characteristic speeds is faster than that of vehicular traffic. This note attempts to provide an overview of the diverging views on how to interpret this property, and specific comments on the interpretation of Helbing and Johansson [On the controversy around Daganzo’s requiem for and Aw-Rascle’s resurrection of second-order traffic flow models, Eur. Phys. J. B (2009), DOI: 10.1140/epjb/e2009-00182-7]. We showed that having faster-than-traffic characteristics does produce counterintuitive predictions, and they cannot be explained away by a linear stability analysis. As such, the existence of such characteristics must be justified by the physics of traffic, and verified through empirical observations.     

Adaptive H∞ synchronization of chaotic systems via linear and nonlinear feedback control

Adaptive H∞ synchronization of chaotic systems via linear and nonlinear feedback control is investigated.The chaotic systems are redesigned by using the generalized Hamiltonian systems and observer approach.Based on Lyapunov’s stability theory,linear and nonlinear feedback control of adaptive H∞ synchronization is established in order to not only guarantee stable synchronization of both master and slave systems but also reduce the effect of external disturbance on an H∞-norm constraint.Adaptive H∞ synchronization of chaotic systems via three kinds of control is investigated with applications to Lorenz and Chen systems.Numerical simulations are also given to identify theeffectiveness of the theoretical analysis.     

EHD instabilities in nematics driven by dichotomous noise: Stability criteria and influence of free boundary conditions

Different criteria for the onset of EHD instabilities in nematic liquid crystals driven by dichotomous stochastic electric fields are compared within a 1d linear model. Sample stability gives always higher, energetic stability always lower thresholds than the first moment's stability investigated in a previous paper, showing the same qualitative behaviour. Especially the direct transition towards chaos above a critical strength of the noise and the change from stabilizing to destabilizing effect of the noise with increasing correlation time can be explained. The influence of free boundary conditions is investigated analyzing first moment's stability of a 2d linear model. The thresholds are slightly higher but behave qualitatively like in the 1d model. The Williams strip pattern becomes more narrow both with increasing strength and mean frequency of the noise.     

Spatiotemporal optical solitons in planar waveguide with periodically modulated cubic-quintic nonlinearity

The propagation and stability of spatiotemporal optical solitons (or optical bullets) in a planar waveguide with periodically modulated cubic-quintic nonlinearity is studied numerically as a function of the amplitudes of modulation (A _m ), the frequency of modulation (ω _m ) and the propagation distance z. The optical spatiotemporal solitons are the result of the balance between the nonlinear parameters, of dispersion (dispersion length, L _D ) and diffraction (diffraction length, L _d ) with temporal and spatial auto-focusing behavior respectively. With the objective of ensure the stability and preventing the collapse or the spreading of pulses, in this study we explore the cubic-quintic nonlinearity with the optical fields coupled by XPM and take into account several values for the non linear parameter α and for amplitudes (A _m ) and frequency (ω _m ) of modulation as a function of the propagation distance z and we cause the collisions of two pulses (envelope of the optical field) to ensure that the optical pulse are solitons. After numerical analysis of parameter settings selected four conditions and for all we get stable solitons and this paper shown that, for a fixed amplitude and frequency of modulation we have stable spatiotemporal solitons.     

Bifurcation and stability of an improved time-delayed fluid flow model in internet congestion control

Based on the fluid flow time-delayed model proposed by Misra et al in internet congestion control, one modified time-delayed model is presented, where the influence of the communication delay on the router queue length is investigated in detail. The main advantage of the new model is that its stability domain is larger even without an extra controller. By linear stability analysis and numerical simulation, the effectiveness and feasibility of the novel model in internet congestion control are verified.     

Irregular Fluctuations in Uncoupled Map Lattices

Analytic approximations for the spatial average and its variance are derived for a system of N uncoupled chaotic logistic maps with growth parameter r = 4. The arising nontrivial closure problem is investigated with various techniques related to the classical moment problem. A Lyapunov-like linear stability analysis is presented for the transient as well as for the fluctuation regime.     

Importance of direction of vibration on the onset of Soret-driven convection under gravity or weightlessness

This paper considers the influence of the direction of vibration on the stability threshold of two-dimensional Soret-driven convection. The configuration is an infinite layer filled with a binary mixture, which can be heated from below or from above. The limiting case of high-frequency and small-amplitude vibration is considered for which the time-averaged formulation has been adopted. The linear stability analysis of the quasi-mechanical equilibrium shows that the problem depends on five non-dimensional parameters. These include the thermal Rayleigh number ( Ra_T), the vibrational parameter (R), the Prandtl number ( Pr), the Lewis number (Le), the separation ratio (S) and the orientation of vibration with respect to the horizontal heated plate (). For different sets of parameters, the bifurcation diagrams are plotted Ra_c = f (S) and k_c = g(S), which are the critical thermal Rayleigh and the critical wave numbers, respectively. Our results indicate that, relative to the classical case of static gravity, vibration may affect all regions in Ra_c-S stability diagram. In the case of mono-cellular convection, by using a regular perturbation method, a closed-form relation for the critical Rayleigh number is found. Several physical situations in the presence or in the absence of gravity (micro-gravity) are discussed.     

Exploration of large amplitude motions in the Ca+Ar2 complex

ABSTRACT

We present a theoretical study of the ground electronic state potential of the Ca⁺Ar₂ complex and of its photoabsorption spectra, simulated at temperatures ranging between 20 and 220?K. These calculations exploit a Monte-Carlo (MC) method, based on a one-electron pseudo-potential approach. A pairwise additive potential fitted to coupled cluster ab initio points, is used to model the Ca⁺Ar₂ complex. Our study shows that the most stable form of Ca⁺Ar₂ is a bent C_2v structure, whereas the linear isomer is located at around 90?±?10?cm^?1 above in energy. The analysis of the photoabsorption spectra establishes that a structural transition from bent Ca⁺Ar₂ to linear ArCa⁺Ar occurs at T～100?K. Trends in binding energies of both isomers, bond lengths and bond angles are also discussed. Molecular orbital overlaps provide an explanation for the order of stability between the bent and linear structures.